Intersection of a Line and a Circle (Part 1): Examples

This lesson will cover a few examples to illustrate the concepts covered in the previous lesson.

Example 1 Determine whether the given line intersects the given circle at two distinct points, touch the circle or does not intersect the circle at any point:

(i) L: 3x + 4y = 10; C: x2­­ + y2 = 9

(ii) L: 5x + 12y = 9; C: x2 + y2 – 4x – 2y + 4 = 0

(iii) L: x = 3; C: x2 + y2 + 4x + 6y – 3 = 0

Solution The problems are quite simple, as they are directly based on what was previously discussed.

(i) I’ll illustrate the first method in this part, i.e. forming the quadratic and using its discriminant. The next two will be solved by using the second method. Let’s begin.

Substituting the value of ‘y’ from the line into the circle’s equation, we get something like x2 + (10 – 3x)2/16 = 9, or 25x2 – 60x – 44 = 0. The discriminant of the equation equals 8000 which is positive. Hence, the line will intersect the circle at two distinct points.

(ii) In this part, we’ll find the distance of the given line from the circle’s center (2, 1) and compare it with its radius, which is 1.

Now the distance of the line from the point (2, 1) is given by |5(2) + 12 – 9|/13 which equals 1, i.e. equal to the radius.

Therefore the given line will touch the circle. Can you find the point where the line touches the circle?

(iii) Again, we’ll find the distance between the given line x = 3, and the center (–2, –3), which equals 5. The radius of the circle is 4, which is smaller than the calculated distance. Hence the line will not intersect or touch the circle.